Hyperbolic volume and Heegaard distance

TL;DR

This paper establishes a bound linking the volume of hyperbolic 3-manifolds and the Heegaard distance of their surfaces, showing larger genus surfaces have bounded distance related to volume.

Contribution

It proves a new relationship between hyperbolic volume and Heegaard distance, introducing bounds based on volume and triangulation complexity.

Findings

Heegaard surfaces of genus greater than a constant times volume have Heegaard distance at most 2.

A bound on Heegaard distance is established for manifolds with bounded tetrahedral triangulations.

The results connect geometric volume with topological complexity in 3-manifolds.

Abstract

We prove (Theorem~1.5) that there exists a constant $\Lambda > 0$ so that if $M$ is a $(\mu,d)$-generic complete hyperbolic 3-manifold of volume $\vol[M] < \infty$ and $\Sigma \subset M$ is a Heegaard surface of genus $g(\Sigma) > \Lambda \vol[M]$, then $d(\Sigma) \leq 2$, where $d(\Sigma)$ denotes the distance of $\Sigma$ as defined by Hempel. The key for the proof of the main result is Theorem~1.8 which is on independent interest. There we prove that if $M$ is a compact 3-manifold that can be triangulated using at most $t$ tetrahedra (possibly with missing or truncated vertices), and $\Sigma$ is a Heegaard surface for $M$ with $g(\Sigma) \geq 76t+26$, then $d(\Sigma) \leq 2$.